Gaza militants fired fresh rocket barrages at Israel early Sunday in a deadly escalation that has seen Israel respond with waves of strikes as a fragile truce again faltered and a further escalation was feared.

Gazan authorities reported four Palestinians killed in Israeli strikes in the escalation that began Saturday, but Israel disputed their account of the deaths of a pregnant mother and her baby.

One 58-year-old Israeli man was killed by a rocket strike on the city of Ashkelon near the Gaza border, Israeli police and the hospital said.

The latest flare-up came with Hamas, the Islamist movement that rules the blockaded Gaza Strip, seeking further concessions from Israel under the ceasefire.

Israel said 430 rockets had been fired from the Palestinian enclave since Saturday and its air defences intercepted many of them.

Beyond the man killed, an 80-year-old woman was seriously injured in a rocket strike on the Israeli city of Kiryat Gat, police and medics said.

A man was also hospitalised in Ashkelon, said police, which spoke of other injuries without providing details.

A house near Ashkelon was damaged while other rockets hit open areas.

The Israeli army said its tanks and planes hit some 200 militant targets in Gaza in response.

They included an Islamic Jihad attack tunnel that stretched from southern Gaza into Israeli territory, it said.

Two multi-storey buildings in Gaza City were destroyed.

Israel said one of the buildings included Hamas military intelligence and security offices.

Turkey said an office for its state news agency Anadolu was located in the building and strongly denounced the strike.

Israel said the other building housed Hamas and Islamic Jihad offices.

The Gaza health ministry said Israeli strikes killed a 14-month-old baby and her pregnant mother, 37, in addition to two Palestinian men, while 40 were wounded.

Israeli army spokesman Avichay Adraee on Twitter challenged the account of the mother and her baby being killed in an Israeli strike, suggesting they died from Palestinian fire.

Adraee did not provide more details and the army refused to comment further. Israel's military said it was targeting only military-related sites.

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As the exchange of fire continued, Israeli Prime Minister Benjamin Netanyahu held consultations with security chiefs on Saturday but had not commented publicly.

A statement from Hamas ally Islamic Jihad claimed responsibility for at least some of the rocket fire and said it was prepared for more.